Activities This Week

I will begin with an introduction to algebraic K-theory, highlighting its connections to number theory and algebraic geometry, with a focus on descent phenomena. Next, I will provide an overview of chromatic homotopy theory and its relationship to K-theory. I will then discuss the redshift conjecture and several significant advances from recent years. Finally, I will present joint work with Carmeli, Schlank and Yanovski on the compatibility of K-theory with chromatic analogues of cyclotomic extensions. I will explain how this result contributed to the recent refutation of Ravenel’s long-standing telescope conjecture.

Given a dynamical system which admits chaos, its possible equilibria are studied through the collection of its invariant measures (i.e the probability of an event does change under the time-evolution). This collection may be very large, and we often wish to single out measures of importance. In particular, physical measures are a sought-after object, as they describe an observable equilibrium. In this talk we will define what are physical measures (in a broad sense, including SRB measure), and list a few recent results and open questions regarding their existence for smooth dynamical systems. In particular, finding “testable” conditions for the existence of physical measures is an on-going and active field of research.